# Sh:464

- Baldwin, J. T., Laskowski, M. C., & Shelah, S. (1993).
*Forcing isomorphism*. J. Symbolic Logic,**58**(4), 1291–1301. arXiv: math/9301208 DOI: 10.2307/2275144 MR: 1253923 -
Abstract:

A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is classifiable if it is superstable and does not have either the dimensional order property or the omitting types order property. Shelah [Sh:c] showed that if a theory T is classifiable then each model of cardinality \lambda is described by a sentence of L_{\infty,\lambda}. In fact this sentence can be chosen in the L^*_{\lambda}. (L^*_{\lambda} is the result of enriching the language L_{\infty,\beth^+} by adding for each \mu<\lambda a quantifier saying the dimension of a dependence structure is greater than \mu.) The truth of such sentences will be preserved by any forcing that does not collapse cardinals \leq\lambda and that adds no new countable subsets of \lambda. Hence, if two models of a classifiable theory of power \lambda are non-isomorphic, they are non-isomorphic after a \lambda-complete forcing. Here we show that the hypothesis of the forcing adding no new countable subsets of \lambda cannot be eliminated. In particular, we show that non-isomorphism of models of a classifiable theory need not be preserved by ccc forcings. - published version (12p)

Bib entry

@article{Sh:464, author = {Baldwin, John T. and Laskowski, Michael Chris and Shelah, Saharon}, title = {{Forcing isomorphism}}, journal = {J. Symbolic Logic}, fjournal = {The Journal of Symbolic Logic}, volume = {58}, number = {4}, year = {1993}, pages = {1291--1301}, issn = {0022-4812}, doi = {10.2307/2275144}, mrclass = {03C45 (03C55)}, mrnumber = {1253923}, mrreviewer = {Bruno Poizat}, doi = {10.2307/2275144}, note = {\href{https://arxiv.org/abs/math/9301208}{arXiv: math/9301208}}, arxiv_number = {math/9301208} }